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Chapter 3: Problem 14

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$5^{2-x}=\frac{1}{125}$$

Short Answer

Expert verified
\The solution to the exponential equation is \(x=5\).

Step by step solution

01

Express the denominator as a power of the base

The given equation is \(5^{2-x}=\frac{1}{125}\). Using the base exponent rule, we know \(5^{-3} = \frac{1}{125}\) so it can be rewritten as \(5^{2-x}=5^{-3}\)
02

Equate the exponents

According to the power rule, if the bases are equal, the exponents are also equal. Therefore, equate the powers and we have \(2-x=-3\)
03

Solve for x

Solving the equation \(2-x=-3\) for \(x\) gives \(x=2+3=5\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponent Rules
Understanding exponent rules is fundamental when dealing with exponential equations. Exponents, or powers, are a shorthand way to express repeated multiplication of a number by itself. Let's go through the basic rules:
  • Product Rule: When multiplying two powers with the same base, you add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
  • Quotient Rule: When dividing two powers with the same base, subtract the exponents. For example, \(\frac{a^m}{a^n} = a^{m-n}\).
  • Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, \( (a^m)^n = a^{m \times n}\).
  • Negative Exponent Rule: A negative exponent represents the reciprocal of the base raised to the opposite positive exponent. For instance, \(a^{-n} = \frac{1}{a^n}\).
  • Zero Exponent Rule: Any non-zero base raised to the zero power is equal to one, so \(a^0 = 1\).
Practicing these rules helps simplify and solve exponential equations effectively.
Equating Exponents
When two expressions with the same base are equal, their exponents must also be equal. This process is known as 'equating exponents.' It's a powerful tool for solving exponential equations. Here's the straightforward process:
  • Rewrite both sides of the equation so that the bases are the same.
  • Since the bases are equal, set the exponents equal to one another.
  • Solve the resulting equation to find the value of the unknown variable.
Applying the principle of equating exponents allows us to deduce that in the equation \(a^m = a^n\), \(m\) must equal \(n\) as long as \(a\) is not zero.
Expressing Numbers as Powers
Many exponential equations can be simplified by expressing numbers as powers of a common base. This strategy is connected to our understanding of exponent rules. To express a number as a power, you need to:
  • Identify a base that can be raised to some power to reach that number.
  • Determine the exponent that makes the base reach the number.
For example, recognizing that 125 is a power of 5 (since \(5^3 = 125\)) allows you to rewrite \(125\) as \(5^3\). Similarly, fractions like \(\frac{1}{125}\) can be written as \(5^{-3}\), since they represent the reciprocal of the number 125. By expressing both sides of an equation as powers of the same base, you can use the strategy of equating exponents to solve the equation.
Exponential Equations Step-by-Step Solution
Solving exponential equations can seem daunting, but breaking the process down into a step-by-step solution can make it manageable:
  1. Express each side as a power of the same base: Look for a way to convert all terms into the same base. This may involve expressing numbers as powers, as previously discussed.
  2. Equating exponents: Once you have the same base on both sides, you can set the exponents equal to each other, since the only way for the bases to be equal is for the exponents to be the same as well.
  3. Solve for the unknown: With the exponents set equal to each other, solve the equation just like you would any other linear equation.
By following these steps, you'll unravel the exponential equation to find the value of the unknown variable. This methodical approach can be applied universally to exponential equations, facilitating your understanding and mathematical development.

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Most popular questions from this chapter

Describe the following property using words: \(\log _{b} b^{x}=x\)

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